woptic: Transport Properties with Wannier Functions and Adaptive k-integration

Transcription

1 woptic: Transport Properties with Wannier Functions and Adaptive k-integration Elias Assmann Institute of Solid State Physics, Vienna University of Technology Split,

2 About this presentation On the menu: band-structure calculation with WIEN2k (briefly) maximally localized Wannier functions with wien2wannier and Wannier90 conductivity, thermopower with woptic explanations alternating with sample calculation ask questions It is better to uncover a little than to cover a lot if you want to try your own hand, come to me

3 Anatomy of a calculation WIEN2k optic Schrödinger equation momentum matrix ψ nk (r), ε n (k) ψ nk ψ mk woptic BZ integration σ(ω), σ(0), S wien2wannier k-space overlaps M (k,b) mn = ψ m,k ψ n,k+b Wannier90 max. loc. WF w nr (r) DMFT self-energy Σ(ω)

4 Where to learn WIEN2k

5 Bloch and Wannier S(T ) & σ(ω) MLWF Where to learn WIEN2k Next year, in Nantes Adaptive k-integration Practicalities

6 Anatomy of a calculation WIEN2k optic Schrödinger equation momentum matrix ψ nk (r), ε n (k) ψ nk ψ mk woptic BZ integration σ(ω), σ(0), S wien2wannier k-space overlaps M (k,b) mn = ψ m,k ψ n,k+b Wannier90 max. loc. WF w nr (r) DMFT self-energy Σ(ω)

7 Bloch waves Bloch s theorem Ĥ = 2 + V(r) with V(r + R) V(r) has solutions Ĥ ψ nk = ε n (k) ψ nk with ψ nk (r) = e ikr u nk (r); u n,k (r + R) u n,k (r) and ε n (k + K) ε n (k) (Simultaneous eigenbasis of Ĥ and translation operators T R ) Usual basis for solid-state calculations But for many applications, a localized basis is preferable

8 Fourier transforms of the Bloch waves w R = V (2π) 3 dk e ikr ψ k BZ Transform k R; r is spectator Properties: periodicity w R (r) w 0 (r R) orthonormality w nr w mr = δ nm δ RR from Marzari et al.

9 Gauge freedom ψ nk carries arbitrary phase φ(k) w R = V (2π) 3 dk e i(φ(k) kr) ψ k stronly non-unique (shape, spread) Idea: choose w k := e iφ(k) ψ k for maximum localization of w R BZ from Marzari et al.

10 Gauge freedom ψ nk carries arbitrary phase φ(k) sin x + 1 sin 3x sin 9x 3 9 w R = V (2π) 3 dk e i(φ(k) kr) ψ k BZ stronly non-unique (shape, spread) sin x sin x sin 3x + + sin 49x 49 1 sin 3x + + sin 249x 249 Idea: choose w k := e iφ(k) ψ k for maximum localization of w R Fourier series converges faster for smoother functions: f C p f n const n p [Wikipedia, Gibbs phenomenon ]

11 From bands to WF isolated band isolated group of bands entangled bands E(k) k w k = e iφ(k) ψ k k k [pictures by J. Kuneš]

12 Maximally localized WF [Marzari et al., RMP (2012)] In the multiband case, e iφ(k) unitary matrix [ U + U = 1 ], w nr = e BZdk ikr U (k) mn ψ mk m

13 Maximally localized WF [Marzari et al., RMP (2012)] In the multiband case, e iφ(k) unitary matrix [ U + U = 1 ], w nr = e BZdk ikr U (k) mn ψ mk m Total spread Ω := n ( r 2 n r 2 n) = Ω[U] + ΩI gauge dependent gauge independent

14 Maximally localized WF [Marzari et al., RMP (2012)] In the multiband case, e iφ(k) unitary matrix [ U + U = 1 ], w nr = e BZdk ikr U (k) mn ψ mk m Total spread Ω := n MLWF procedure: minimize Ω[U] (Wannier90) gauge independent ( r 2 n r 2 n) = Ω[U] + ΩI gauge dependent input: M (k,b) mn = ψ mk e ib(k+b) ψ nk optional input: A (k) mn = ψ mk g nk output: U (k) nm, ( ) H (R) nm wien2wannier

15 Anatomy of a calculation WIEN2k optic Schrödinger equation momentum matrix ψ nk (r), ε n (k) ψ nk ψ mk woptic BZ integration σ(ω), σ(0), S wien2wannier k-space overlaps M (k,b) mn = ψ m,k ψ n,k+b Wannier90 max. loc. WF w nr (r) DMFT self-energy Σ(ω)

16 From bands to WF isolated band isolated group of bands entangled bands E(k) k k k w k = e iφ(k) ψ k w nk = U (k) mn ψ mk [pictures by J. Kuneš]

17 Disentanglement Cu (fcc): What to do when #bands > #WF? Ansatz: Select optimally smooth subspace rectangular matrix V k (#bands(k) #WF)! Ω I = Ω I [V] minimize ΩI [V] from Marzari et al. 5 d-like WF, 2 interstitial s-like WF Ω = Ω[U] + Ω I [V]

18 From bands to WF isolated band isolated group of bands entangled bands E(k) k k k w k = e iφ(k) ψ k w nk = U (k) mn ψ mk w nk = U (k) in V(k) mi ψ mk [pictures by J. Kuneš]

19 Anatomy of a calculation WIEN2k optic Schrödinger equation momentum matrix ψ nk (r), ε n (k) ψ nk ψ mk woptic BZ integration σ(ω), σ(0), S wien2wannier k-space overlaps M (k,b) mn = ψ m,k ψ n,k+b Wannier90 max. loc. WF w nr (r) DMFT self-energy Σ(ω)

20 Choice of Wannier subspace SrVO 3 : V-e g V-t 2g O-p V-centered xy orbital

21 Applications of MLWF analysis of chemical bonding electric polarization and orbital magnetization BerryPI Wannier interpolation K G H(k) K F H(R) K 1 Wannier functions as basis functions tight-binding model H(k) = U + (k) ε(k) U(k) realistic dynamical mean-field theory (DMFT) F 1 H(k) G

22 Transport properties with WIEN2k BoltzTrap semi-classical (Boltzmann) band velocities ε(k)/ k instead of momentum matrix elements ψ ψ BoltzWann similar, with Wannier functions woptic quantum-mechanical linear response (Kubo) adaptive BZ integration inclusion of local self-energy Σ(ω) more information: WIEN2k.at reg. users unsupported wien2wannier woptic preprint Wissgott et al., PRB (2012)

23 Calculating optical conductivity and thermopower Very schematically: linear response (Kubo formula): Ĥ = Ĥ0 + λ(t) B t [A(t), Â (t) = Â 0 h i dt λ(t ) B(t ) ] + 0 σ, S: current operators Ψ + Ψ ( Ψ + ) Ψ momentum matrix elements ψ ψ [ĵ(r, χ ret el el (r r, t t ) θ(t t ) t), ĵ(r, t ) ] 0 [ĵ(r, χ ret el heat (r r, t t ) θ(t t ) t), Q(r, t ) ] 0 el. current operator heat-current operator

24 Calculating optical conductivity and thermopower Very schematically: linear response (Kubo formula): Ĥ = Ĥ0 + λ(t) B t [A(t), Â (t) = Â 0 h i dt λ(t ) B(t ) ] + 0 σ, S: current operators Ψ + Ψ ( Ψ + ) Ψ momentum matrix elements ψ ψ [ĵ(r, χ ret el el (r r, t t ) θ(t t ) t), ĵ(r, t ) ] 0 [ĵ(r, χ ret el heat (r r, t t ) θ(t t ) t), Q(r, t ) ] 0 el. current operator heat-current operator

25 Calculating optical conductivity and thermopower Very schematically: linear response (Kubo formula): Ĥ = Ĥ0 + λ(t) B t [A(t), Â (t) = Â 0 h i dt λ(t ) B(t ) ] + 0 σ, S: current operators Ψ + Ψ ( Ψ + ) Ψ momentum matrix elements ψ ψ [ĵ(r, χ ret el el (r r, t t ) θ(t t ) t), ĵ(r, t ) ] 0 [ĵ(r, χ ret el heat (r r, t t ) θ(t t ) t), Q(r, t ) ] 0 el. current operator heat-current operator

26 Adaptive k-integration Al optical conductivity σ(k, ω) ω [ev] σk (ω) [10 6 Ω 1 ev 1 ] W L Γ X W K 2 ω [ev] 0 2

27 Adaptive k-integration Al optical conductivity σ(k, ω) ω [ev] σk (ω) [10 6 Ω 1 ev 1 ] W L Γ X W K 2 ω [ev] 0 2

28 Tetrahedral mesh management tetrahedron T has integration error estimate ɛ T refine T if ɛ T Θ max T ɛ T harshness Θ [0, 1] L Γ X W K enforce regularity at most one hanging node for stable convergence enforce shape stability no heavily-distorted octahedra avoid large integration errors X L Γ W K

29 Tetrahedral mesh management tetrahedron T has integration error estimate ɛ T refine T if ɛ T Θ max T ɛ T harshness Θ [0, 1] enforce regularity at most one hanging node for stable convergence enforce shape stability no heavily-distorted octahedra avoid large integration errors

30 Tetrahedral mesh management tetrahedron T has integration error estimate ɛ T refine T if ɛ T Θ max T ɛ T harshness Θ [0, 1] enforce regularity at most one hanging node for stable convergence enforce shape stability no heavily-distorted octahedra avoid large integration errors Kuhn triangulation

31 Interpolation and random-phase problem DMFT in Wannier basis self-energy Σ i (ω) spectral function A(k, ω) in evaluation of χs: { } tr v(k) A(k) v(k) A(k)

32 Interpolation and random-phase problem DMFT in Wannier basis self-energy Σ i (ω) spectral function A(k, ω) in evaluation of χs: { } tr v(k) A(k) v(k) A(k) from DMFT momentum matrix w nk w mk ; from WIEN2k ψ nk carries random phase φ n (k), which propagates to v but not A problem with w w (v wx A xy v yz A zw ) and w ψ (v wi A ii v ix A xw ) terms (planned) solution: interpolate v(k)

33 Anatomy of a calculation WIEN2k optic Schrödinger equation momentum matrix ψ nk (r), ε n (k) ψ nk ψ mk woptic BZ integration σ(ω), σ(0), S wien2wannier k-space overlaps M (k,b) mn = ψ m,k ψ n,k+b Wannier90 max. loc. WF w nr (r) DMFT self-energy Σ(ω)

34 Limitations Kohn-Sham eigenstates interpreted as excited states scissors operator: ε cond (k) ε LDA cond(k) + careful with doping rigid-band treatment disentanglement not implemented random-phase problem for optical conductivity when using self-energy not-so-mature code

35 Some results for Na x CoO 2 [Wissgott et al., PRB (2010)]

36 Big Thanks to... The code authors As well as Karsten Held Peter Blaha Alessandro Toschi Philipp Wissgott TU Vienna Jan Kuneš Institute of Physics, Prague

37 Big Thanks to... The code authors As well as Karsten Held Peter Blaha Alessandro Toschi Philipp Wissgott TU Vienna Jan Kuneš Institute of Physics, Prague And to my audience for your patience!